Durbin-Watson test
data: modelo.dca
DW = 1.7648, p-value = 0.7037
alternative hypothesis: true autocorrelation is not 0
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, los residuos del rendimiento son completamente aleatorios e independientes.
Normalidad de residuos
\(H_0: \text{La distribución de los residuos del rendimiento es similar a la función normal}\)
\(H_1: \text{La distribución de los residuos del rendimiento es similar a la función normal}\)
shapiro.test(rstudent(modelo.dca))
Shapiro-Wilk normality test
data: rstudent(modelo.dca)
W = 0.94978, p-value = 0.2479
ad.test(rstudent(modelo.dca))
Anderson-Darling normality test
data: rstudent(modelo.dca)
A = 0.51954, p-value = 0.169
lillie.test(rstudent(modelo.dca))
Lilliefors (Kolmogorov-Smirnov) normality test
data: rstudent(modelo.dca)
D = 0.12748, p-value = 0.3684
Asymptotic one-sample Kolmogorov-Smirnov test
data: rstudent(modelo.dca)
D = 0.1198, p-value = 0.8656
alternative hypothesis: two-sided
cvm.test(rstudent(modelo.dca))
Cramer-von Mises normality test
data: rstudent(modelo.dca)
W = 0.090314, p-value = 0.1445
pearson.test(rstudent(modelo.dca))
Pearson chi-square normality test
data: rstudent(modelo.dca)
P = 7.96, p-value = 0.1585
sf.test(rstudent(modelo.dca))
Shapiro-Francia normality test
data: rstudent(modelo.dca)
W = 0.93585, p-value = 0.108
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, la distribución de los residuos del rendimiento es similar a la función normal o gaussiana.
Igualdad de varianzas
data <- data %>%mutate(residuals =residuals(modelo.dca),rstudent =rstudent(modelo.dca))
\(H_1: \text{Al menos en al menos un tratamiento la } \sigma \text{ del rendimiento es diferente a los demás}\)
cochran.test(rdt ~ trt, data = data)
Cochran test for outlying variance
data: rdt ~ trt
C = 0.58356, df = 5, k = 5, p-value = 0.02564
alternative hypothesis: Group B has outlying variance
sample estimates:
A B C D E
8.06633 28.34847 4.97177 4.26752 2.92472
cochranTest(rdt ~ trt, data = data)
Cochran test for outlying variance
data: rdt by trt
C = 0.58356, k = 5, n = 5, p-value = 0.02564
alternative hypothesis: greater
sample estimates:
group var
2.00000 28.34847
hartleyTest(rdt ~ trt, data = data)
Hartley's maximum F-ratio test of homogeneity of variances
data: rdt by trt
F Max = 9.6927, df = 4, k = 5, p-value = 0.2493
bartlett.test(rdt ~ trt, data = data)
Bartlett test of homogeneity of variances
data: rdt by trt
Bartlett's K-squared = 6.5762, df = 4, p-value = 0.16
leveneTest(rdt ~ trt, data = data)
Levene's Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 4 1.3538 0.2851
20
leveneTest(rdt ~ trt, data = data, center ="mean")
Levene's Test for Homogeneity of Variance (center = "mean")
Df F value Pr(>F)
group 4 2.5479 0.0712 .
20
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
fligner.test(rdt ~ trt, data = data)
Fligner-Killeen test of homogeneity of variances
data: rdt by trt
Fligner-Killeen:med chi-squared = 3.8461, df = 4, p-value = 0.4272
Homocedasticidad
\(H_0: \text{La varianza del rendimiento es constante con respecto a los valores ajustados del rendimiento}\)
\(H_1: \text{La varianza del rendimiento no es constante con respecto a los valores ajustados del rendimiento}\)
ncvTest(modelo.dca)
Non-constant Variance Score Test
Variance formula: ~ fitted.values
Chisquare = 2.569732, Df = 1, p = 0.10893
bptest(modelo.dca)
studentized Breusch-Pagan test
data: modelo.dca
BP = 9.1322, df = 4, p-value = 0.05788
bptest(modelo.dca, studentize = F)
Breusch-Pagan test
data: modelo.dca
BP = 11.87, df = 4, p-value = 0.01834
olsrr::ols_test_breusch_pagan(modelo.dca)
Breusch Pagan Test for Heteroskedasticity
-----------------------------------------
Ho: the variance is constant
Ha: the variance is not constant
Data
-------------------------------
Response : rdt
Variables: fitted values of rdt
Test Summary
----------------------------
DF = 1
Chi2 = 2.569732
Prob > Chi2 = 0.1089262
Conclusión. A un nivel de significancia de 0.1, se concluye que existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, la varianza del rendimiento no es constante con respecto a los valores ajustados del rendimiento.
Recomendación. Debido a que no se cumple con el supuesto de homocedasticidad, para evaluar los efectos de los tratamientos con respecto al rendimiento, se debería utilizar pruebas no paramétricas o modelos lineales generalizados a fin cumplir con los supuestos requeridos por las pruebas.
Conclusión. A un nivel de significancia de 0.05, se concluye que existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, al menos un tratamiento tiene un efecto sobre el rendimiento estadísticamente diferente del resto de tratamientos.
Comparaciones de medias
A vs B:
\(H_0: \mu_{A} - \mu_{B} = 0\)
\(H_1: \mu_{A} - \mu_{B} \neq 0\)
A vs C:
\(H_0: \mu_{A} - \mu_{C} = 0\)
\(H_1: \mu_{A} - \mu_{C} \neq 0\)
A vs D:
\(H_0: \mu_{A} - \mu_{D} = 0\)
\(H_1: \mu_{A} - \mu_{D} \neq 0\)
A vs E:
\(H_0: \mu_{A} - \mu_{E} = 0\)
\(H_1: \mu_{A} - \mu_{E} \neq 0\)
B vs C:
\(H_0: \mu_{B} - \mu_{C} = 0\)
\(H_1: \mu_{B} - \mu_{C} \neq 0\)
B vs D:
\(H_0: \mu_{B} - \mu_{D} = 0\)
\(H_1: \mu_{B} - \mu_{D} \neq 0\)
B vs E:
\(H_0: \mu_{B} - \mu_{E} = 0\)
\(H_1: \mu_{B} - \mu_{E} \neq 0\)
C vs D:
\(H_0: \mu_{C} - \mu_{D} = 0\)
\(H_1: \mu_{C} - \mu_{D} \neq 0\)
C vs E:
\(H_0: \mu_{C} - \mu_{E} = 0\)
\(H_1: \mu_{C} - \mu_{E} \neq 0\)
D vs E:
\(H_0: \mu_{D} - \mu_{E} = 0\)
\(H_1: \mu_{D} - \mu_{E} \neq 0\)
Prueba de HSD (Honestamente significativa -> Tukey)
agricolae::HSD.test(modelo.dca, trt ="trt", alpha =0.05,group =TRUE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
HSD Test for rdt
Mean Square Error: 9.715762
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 20
Critical Value of Studentized Range: 4.231857
Minimun Significant Difference: 5.899081
Treatments with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 ab
B 51.718 bc
A 47.134 c
agricolae::HSD.test(modelo.dca, trt ="trt", alpha =0.05,group =FALSE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
HSD Test for rdt
Mean Square Error: 9.715762
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 20
Critical Value of Studentized Range: 4.231857
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.1780 -10.483081 1.315081
A - C -8.594 0.0025 ** -14.493081 -2.694919
A - D -12.034 0.0001 *** -17.933081 -6.134919
A - E -11.744 0.0001 *** -17.643081 -5.844919
B - C -4.010 0.2864 -9.909081 1.889081
B - D -7.450 0.0092 ** -13.349081 -1.550919
B - E -7.160 0.0128 * -13.059081 -1.260919
C - D -3.440 0.4308 -9.339081 2.459081
C - E -3.150 0.5155 -9.049081 2.749081
D - E 0.290 0.9999 -5.609081 6.189081
Nota: Todos aquellos tratamientos que compartan por lo menos una letra serán estadísticamente similares.
Recomendación: Se debe usar la prueba de Tukey cuando como máximo se tenga 6 tratamientos.
Prueba de Duncan
agricolae::duncan.test(modelo.dca, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
Duncan's new multiple range test
for rdt
Mean Square Error: 9.715762
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 20
Critical Range
2 3 4 5
4.112208 4.316437 4.446229 4.536881
Means with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 ab
B 51.718 b
A 47.134 c
agricolae::duncan.test(modelo.dca, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
Duncan's new multiple range test
for rdt
Mean Square Error: 9.715762
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0307 * -8.696208 -0.4717917
A - C -8.594 0.0004 *** -12.910437 -4.2775629
A - D -12.034 0.0000 *** -16.570881 -7.4971186
A - E -11.744 0.0000 *** -16.190229 -7.2977710
B - C -4.010 0.0554 . -8.122208 0.1022083
B - D -7.450 0.0020 ** -11.896229 -3.0037710
B - E -7.160 0.0023 ** -11.476437 -2.8435629
C - D -3.440 0.1132 -7.756437 0.8764371
C - E -3.150 0.1258 -7.262208 0.9622083
D - E 0.290 0.8845 -3.822208 4.4022083
Nota: La prueba de Duncan usa múltiples valores críticos.
Recomendación: No usar la prueba de Duncan cuando se presentan resultados con múltiples tratamientos que obtengan varias significancias.
Prueba de Student - Newman -Keuls
agricolae::SNK.test(modelo.dca, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
Student Newman Keuls Test
for rdt
Mean Square Error: 9.715762
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 20
Critical Range
2 3 4 5
4.112208 4.987534 5.517742 5.899081
Means with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 ab
B 51.718 b
A 47.134 c
agricolae::SNK.test(modelo.dca, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
Student Newman Keuls Test
for rdt
Mean Square Error: 9.715762
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0307 * -8.696208 -0.4717917
A - C -8.594 0.0008 *** -13.581534 -3.6064665
A - D -12.034 0.0001 *** -17.933081 -6.1349187
A - E -11.744 0.0000 *** -17.261742 -6.2262576
B - C -4.010 0.0554 . -8.122208 0.1022083
B - D -7.450 0.0060 ** -12.967742 -1.9322576
B - E -7.160 0.0045 ** -12.147534 -2.1724665
C - D -3.440 0.2136 -8.427534 1.5475335
C - E -3.150 0.1258 -7.262208 0.9622083
D - E 0.290 0.8845 -3.822208 4.4022083
Prueba de Least significant difference
agricolae::LSD.test(modelo.dca, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
LSD t Test for rdt
Mean Square Error: 9.715762
trt, means and individual ( 95 %) CI
rdt std r LCL UCL Min Max
A 47.134 2.840129 5 44.22623 50.04177 42.26 49.43
B 51.718 5.324328 5 48.81023 54.62577 44.41 57.31
C 55.728 2.229747 5 52.82023 58.63577 53.72 59.45
D 59.168 2.065798 5 56.26023 62.07577 55.87 60.89
E 58.878 1.710181 5 55.97023 61.78577 55.87 60.17
Alpha: 0.05 ; DF Error: 20
Critical Value of t: 2.085963
least Significant Difference: 4.112208
Treatments with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 ab
B 51.718 b
A 47.134 c
agricolae::LSD.test(modelo.dca, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dca ~ "trt"
LSD t Test for rdt
Mean Square Error: 9.715762
trt, means and individual ( 95 %) CI
rdt std r LCL UCL Min Max
A 47.134 2.840129 5 44.22623 50.04177 42.26 49.43
B 51.718 5.324328 5 48.81023 54.62577 44.41 57.31
C 55.728 2.229747 5 52.82023 58.63577 53.72 59.45
D 59.168 2.065798 5 56.26023 62.07577 55.87 60.89
E 58.878 1.710181 5 55.97023 61.78577 55.87 60.17
Alpha: 0.05 ; DF Error: 20
Critical Value of t: 2.085963
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0307 * -8.696208 -0.4717915
A - C -8.594 0.0003 *** -12.706208 -4.4817915
A - D -12.034 0.0000 *** -16.146208 -7.9217915
A - E -11.744 0.0000 *** -15.856208 -7.6317915
B - C -4.010 0.0554 . -8.122208 0.1022085
B - D -7.450 0.0012 ** -11.562208 -3.3377915
B - E -7.160 0.0017 ** -11.272208 -3.0477915
C - D -3.440 0.0963 . -7.552208 0.6722085
C - E -3.150 0.1258 -7.262208 0.9622085
D - E 0.290 0.8845 -3.822208 4.4022085